Is a ratio of two unbiased averages unbiased?

Question

Two independent samples give unbiased averages: $\bar X$ for the mean cost $\mu_X$ and $\bar Y$ for the mean revenue $\mu_Y > 0$ . You estimate the ratio $r = \mu_X/\mu_Y$ with the plug-in ratio $\hat r = \bar X / \bar Y$ .

Is $\hat r$ an unbiased estimator of $r$ ? Explain why, and describe the size of any bias.

Accepted Answer

A ratio is a nonlinear function. Even though \mathbb{E}[\bar X] = \muX and \mathbb{E}[\bar Y] = \muY, the ratio \bar X/\bar Y is nonlinear in \bar Y. Because the two samples are independent, \mathbb{E}[\hat r] = \mathbb{E}[\bar X]\cdot\mathbb{E}\!\left[\frac{1}{\bar Y}\right] = \muX\,\mathbb{E}\!\left[\frac{1}{\bar Y}\right]. The estimator would be unbiased only if \mathbb{E}[1/\bar Y] = 1/\muY, but 1/y is strictly convex, so by Jensen \mathbb{E}[1/\bar Y] 1/\muY. Hence \hat r is biased (high when \muX 0). The size of the bias. A second-order expansion gives \mathbb{E}[\hat r] \approx \frac{\…

Is a ratio of two unbiased averages unbiased?

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